Theorem ltexnq 10890

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10841	. . . 4 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5690	. . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3		ordpinq 10858	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
4		elpqn 10840	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ (N × N))
6		xp1st 7967	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐴) ∈ N)
8		elpqn 10840	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ (N × N))
10		xp2nd 7968	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐵) ∈ N)
12		mulclpi 10808	. . . . . . 7 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
13	7, 11, 12	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
14		xp1st 7967	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
15	9, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐵) ∈ N)
16		xp2nd 7968	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐴) ∈ N)
18		mulclpi 10808	. . . . . . 7 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
19	15, 17, 18	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
20		ltexpi 10817	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
21	13, 19, 20	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
22		relxp 5643	. . . . . . . . . . . 12 ⊢ Rel (N × N)
23	4	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 ∈ (N × N))
24		1st2nd 7985	. . . . . . . . . . . 12 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	22, 23, 24	sylancr 588	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25	oveq1d 7375	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
27	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐴) ∈ N)
28	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐴) ∈ N)
29		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝑦 ∈ N)
30		mulclpi 10808	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
31	17, 11, 30	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33		addpipq 10852	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (𝑦 ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
34	27, 28, 29, 32, 33	syl22anc 839	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
35	26, 34	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
36		oveq2 7368	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37		distrpi 10813	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦))
38		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐴) ∈ V
39		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐴) ∈ V
40		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐵) ∈ V
41		mulcompi 10811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42		mulasspi 10812	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
43	38, 39, 40, 41, 42	caov12 7588	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = ((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
44		mulcompi 10811	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N 𝑦) = (𝑦 ·N (2nd ‘𝐴))
45	43, 44	oveq12i 7372	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦)) = (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴)))
46	37, 45	eqtr2i 2761	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦))
47		mulasspi 10812	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵)))
48		mulcompi 10811	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))
49	48	oveq2i 7371	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵))) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
50	47, 49	eqtri 2760	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
51	36, 46, 50	3eqtr4g 2797	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)))
52		mulasspi 10812	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
53	52	eqcomi 2746	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)))
55	51, 54	opeq12d 4838	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩)
56	55	eqeq2d 2748	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ ↔ (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
57	35, 56	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
58		fveq2 6835	. . . . . . . . 9 ⊢ ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
59		adderpq 10871	. . . . . . . . . . 11 ⊢ (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
60		nqerid 10848	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
61	60	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐴) = 𝐴)
62	61	oveq1d 7375	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
63	59, 62	eqtr3id 2786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
64		mulclpi 10808	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
65	17, 17, 64	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
67	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐵) ∈ N)
68	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐵) ∈ N)
69		mulcanenq 10875	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
70	66, 67, 68, 69	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
71	8	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 ∈ (N × N))
72		1st2nd 7985	. . . . . . . . . . . . . 14 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
73	22, 71, 72	sylancr 588	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
74	70, 73	breqtrrd 5127	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵)
75		mulclpi 10808	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
76	66, 67, 75	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
77		mulclpi 10808	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (2nd ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
78	66, 68, 77	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
79	76, 78	opelxpd 5664	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
80		nqereq 10850	. . . . . . . . . . . . 13 ⊢ ((⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
81	79, 71, 80	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
82	74, 81	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵))
83		nqerid 10848	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Q → ([Q]‘𝐵) = 𝐵)
84	83	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐵) = 𝐵)
85	82, 84	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = 𝐵)
86	63, 85	eqeq12d 2753	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
87	58, 86	imbitrid 244	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
88	57, 87	syld 47	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
89		fvex 6848	. . . . . . . 8 ⊢ ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) ∈ V
90		oveq2 7368	. . . . . . . . 9 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
91	90	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
92	89, 91	spcev 3561	. . . . . . 7 ⊢ ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
93	88, 92	syl6 35	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
94	93	rexlimdva 3138	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
95	21, 94	sylbid 240	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
96	3, 95	sylbid 240	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
97	2, 96	mpcom 38	. 2 ⊢ (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
98		eleq1 2825	. . . . . . 7 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q ↔ 𝐵 ∈ Q))
99	98	biimparc 479	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
100		addnqf 10863	. . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q
101	100	fdmi 6674	. . . . . . 7 ⊢ dom +Q = (Q × Q)
102		0nnq 10839	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
103	101, 102	ndmovrcl 7546	. . . . . 6 ⊢ ((𝐴 +Q 𝑥) ∈ Q → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
104		ltaddnq 10889	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
105	99, 103, 104	3syl 18	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
106		simpr 484	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
107	105, 106	breqtrd 5125	. . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
108	107	ex 412	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
109	108	exlimdv 1935	. 2 ⊢ (𝐵 ∈ Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
110	97, 109	impbid2 226	1 ⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  ⟨cop 4587   class class class wbr 5099   × cxp 5623  Rel wrel 5630  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Ncnpi 10759   +_N cpli 10760   ·_N cmi 10761   <_N clti 10762   +_pQ cplpq 10763   ~_Q ceq 10766  Qcnq 10767  [Q]cerq 10769   +_Q cplq 10770   <_Q cltq 10773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-omul 8404  df-er 8637  df-ni 10787  df-pli 10788  df-mi 10789  df-lti 10790  df-plpq 10823  df-mpq 10824  df-ltpq 10825  df-enq 10826  df-nq 10827  df-erq 10828  df-plq 10829  df-mq 10830  df-1nq 10831  df-ltnq 10833

This theorem is referenced by:  ltbtwnnq  10893  prnmadd  10912  ltexprlem4  10954  ltexprlem7  10957  prlem936  10962